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Superstring Field Theory, Superforms and Supergeometry

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Superstring Field Theory, Superforms and Supergeometry DISIT-2018 ARC-18-05 SUPERSTRING FIELD THEORY, SUPERFORMS AND SUPERGEOMETRY ROBERTO CATENACCI, PIETRO ANTONIO GRASSI, AND SIMONE NOJA Abstract. Inspired by superstring field theory, we study differential, integral, and inverse forms and their mutual relations on a supermanifold from a sheaf-theoretical point of view. In particular, the formal distributional properties of integral forms are recovered in this scenario in a geometrical way. Further, we show how inverse forms \extend" the ordinary de Rham complex on a supermanifold, thus providing a mathematical foundation of the Large Hilbert Space used in superstrings. Last, we briefly discuss how the Hodge diamond of a supermanifold looks like, and we explicitly compute it for super Riemann surfaces. Contents 1. Introduction 1 2. The Large Hilbert Space, PCO's and New Superforms 3 3. Elements of Supermanifolds 8 4. Locally-Free Sheaves on Supermanifolds: Tangent, Cotangent and Berezinian Sheaves 9 5. Superforms and Integral Forms Complex on a Supermanifold 16 6. Negative Degree Superforms and their Complex 21 6.1. Large Hilbert Space and Cechˇ Cohomology 25 6.2. Large Hilbert Space and Calabi-Yau Supermanifolds 27 7. Superforms and Pseudo-forms for Higher Odd Dimensions 29 8. Serre Duality and Hodge Diamond of a Supermanifold 33 9. Conclusions and Outlooks 38 Acknowledgements 39 References 39 1. Introduction Supergeometry is a fascinating branch of mathematics that prompted from the physical motivation of describing fermionic degrees of freedom. As is well known since the first years of quantum mechanics, identical particles can appear in two types: bosons and fermions. They have different properties, but essentially their wave functions, describing the states of those particles, have to be either fully symmetrized under the exchange of two identical particles in the case of bosons, or fully anti-symmetrized in the case of fermions. Such a requirement is easily implemented by representing the fermions in terms of anticommuting variables, also said Grassmann variables belonging to a superalgebra. This original motivation stemming from physics has given a strong impulse to the study of supergeometry, a context in which commuting and anticommuting variables can be treated on the same footing and described in a 1 2 ROBERTO CATENACCI, PIETRO ANTONIO GRASSI, AND SIMONE NOJA unified fashion. Nonetheless, further important developments were motivated by string theory and string field theory. In string theory, in order to include the fermionic physical degrees of freedom and also protect- ing the theory from unwanted tachyonic fields and stabilising the vacuum, one needs fermionic coordinates (either in the vector representation of Lorentz group, RNS formulation, or in the spinor representation, GS/pure spinor formulation). In this respect, the spacetime is enriched by these additional coordinates and the supergeometry starts playing a fundamental role. On one side, string theory needs the supergeometry formulation to define vertex operators, corre- lation functions and amplitudes, on the other side the geometry emerging from that embodies those anticommuting variables in the properties of supermanifolds. During the last years, several research articles [1, 2] pointed out new important applications of supergeometry in the context of string theory. In particular, it has been observed that the correlation functions of vertex operators, after integrating over conformal fields, are special types of differential forms - known as integral forms - on the supermoduli space of super Riemann surfaces. To complete the computation, one needs an integration on that supermoduli space, which proved to be a formidable hard problem as one has to confront with some typical supergeometric subtleties, as recently shown by Donagi and Witten in [3]. By the way, this kind of issues called for the definition of an integration theory on supermanifolds. This has been developed and it revealed new interesting features of differential forms: 1) the differential forms on a supermanifold are characterized by two numbers: the form degree and the picture number, 2) the complex of superforms must be extended to integral forms. This is obtained by adding to the complex additional lines with fixed picture and variable form degree. 3) In general, picture-zero differential forms have no upper bound to their form degree, whilst integral forms, i.e. those forms having maximal picture number, have no lower bound to their form degree (which can also be negative). Finally, differential forms with a generic picture number are unbounded from above and from below. In addition, at a given form number the forms with a non-maximal picture number span an infinite-dimensional space. 4) New differential operators can be defined in order to remove or to add picture to the differential forms. All these features are easily discussed in the context of conformal field theory where the calculations can be performed. Nevertheless, some of the computations have a geometrical origin and therefore these features can be translated in term of geometrical properties. For that purpose, we use a sheaf-theoretical approach to supermanifolds. Nonetheless, to keep our exposition as readable and concrete as possible, we will use as prototypical example for our considerations and constructions the projective superspaces Pnjm, whose supermanifold structure is non-trivial but easy-enough to allows us for explicit computations in order to identify the sheaves involved and make clear their sheaf-theoretical local-to-global nature. Also, some of the computational properties of integral forms are to be ascribed to their distributional nature and therefore it is shown how analytical distributional properties and geometrical aspects fuse into a precise description. This also motivates the introduction of a new type of superforms, called negative-degree superforms or inverse forms, which have interesting properties. They play an essential role in the comparison between string theory and supergeometry. Indeed, in the string theory framework it is known how to enlarge the physical spectrum of states (called Large Hilbert Space) in order to gain a useful description of the BRST cohomology (vertex operator observables): in this paper we will show how this is achieved from a purely geometrical approach, shedding some light on the supergeometrical origin of concepts underlying string field theory. The Large Hilbert Space has new features that have never been considered in supergeometry revealing new interesting results. 3 The main motivations of the present work is the translation into a rigorous mathematical framework of the the properties of differential superforms, integral forms and inverse forms via sheaf theory. A future goal is to understand if the A1-algebra appearing in super string field theory could show up also in the supergeometric context, possibly in a natural fashion. The plan of the paper is the following: in sec. 2, we revise some ingredients from physical perspective such as the beta-gamma ghost fields, their fermionization, their vertex operators and their OPE algebras. In addition, we recall some basics facts regarding distributions and how they have to be understood in the present context; finally, picture changing operator are preliminarily discussed here. In sec. 3 and sec. 4, we recall basic facts about supermanifolds and we introduce some of the natural sheaves (namely the tangent, the cotangent and the Berezinian sheaves) that can be defined over a supermanifold and that will enter our description. In sec. 5, we introduce a global definition of the sheaves of integral forms and related complex. In sec. 6, we introduce the new concept of negative-degree superforms (a.k.a. inverse forms) and their complex and we discuss the cohomology of Large Hilbert Space in two interesting instances. In sec. 7, some issues in higher odd dimensions are addressed and discussed. Finally, in sec. 8, using mostly Serre duality, we briefly address the problem of attaching a Hodge diamond to a complex supermanifold, by underlying the differences arising in comparison with the ordinary well-understood case: the relevant case of super Riemann surfaces is described in some details. 2. The Large Hilbert Space, PCO's and New Superforms The ideas of the Large Hilbert Space (LHS) and of the Picture Changing Operators (PCO) have been introduced in string theory [4], in order to quantize the ghost fields associated to the superdiffeomorphisms on the worldsheet. Nonetheless those ideas can be imported in the geometry of supermanifolds and, as will be shown, lead to new interesting addition to the space of integral forms. In particular, it will be shown that the space of distributions such as the Dirac delta forms (local expressions for integral forms), used so far as prototypes is not large enough and it must be augmented to the full set of distributional forms with compact support. In the quantization of superstring theory (see [5] for a comprehensive and complete review using the notation of the present section), one introduces two sets of conformal fields with con- formal weights (2; −1) and (3=2; −1=2) needed to fix the local supersymmetry and worldsheet diffeomorphisms. They are named ghost and superghost fields and denoted by (b(z); c(z)) and (β(z); γ(z)), respectively. The first set is made of anticommuting fields, while the second one by commuting real fields. The quantization of the latter requires some additional care since any function of the zero mode of γ enters in the cohomology. Such a degree of freedom has the same properties of the differential dθ of the worldsheet anticommuting local coordinate θ of the super Riemann surface in the local coordinate system (z; θ). A powerful way to deal with the quantization of these fields is by performing a fermionization (see [4]) by expressing the set (β(z); γ(z)) in terms of two anticommuting fields (ξ(z); η(z)) (with conformal weight (0; 1)) and one chiral boson φ(z) as follows γ(z) = : η(z) eφ(z):; β(z) = : @ξ(z) e−φ(z) :; δ(γ(z)) = : e−φ(z) :; δ(β(z)) = : eφ(z) :; (2.1) The colon notation, as usual, denotes the normal ordering in the products.
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